Let Vr done the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2 r-1). Let Tr=Vr+1-Vr-2 and Qr=Tr+1-Tr for r=1,2, ldots Tr is always

Question

Let Vr done the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2 r-1). Let Tr=Vr+1-Vr-2 and Qr=Tr+1-Tr for r=1,2, ldots Tr is always (A) an odd number. (B) an even number. (C) a prime number. (D) a composite number.

Tardigrade Admin · Accepted Answer

We have vr=(1/2)(2 r3-r2+r) vr+1=(1/2)[2(r+1)3-(r+1)2+(r+1)] Tr=vr+1-vr =(r+1)3-(1/2)[(r+1)2-r2]+(1/2)(1) =2 r2+2 r-1 =(r+1)(3 r-1) which is a composite number.

Q. Let $V_{r}$ done the sum of the first $r$ terms of an arithmetic progression (A.P.) whose first term is $r$ and the common difference is $(2 r - 1)$ . Let $T_{r} = V_{r + 1} - V_{r} - 2$ and $Q_{r} = T_{r + 1} - T_{r}$ for $r = 1, 2, \dots$
$T_{r}$ is always

an odd number.

an even number.

a prime number.

a composite number.

We have
$v_{r} = \frac{1}{2} (2 r^{3} - r^{2} + r)$
$v_{r + 1} = \frac{1}{2} [2 (r + 1)^{3} - (r + 1)^{2} + (r + 1)]$
$T_{r} = v_{r + 1} - v_{r}$
$= (r + 1)^{3} - \frac{1}{2} [(r + 1)^{2} - r^{2}] + \frac{1}{2} (1)$
$= 2 r^{2} + 2 r - 1$
$= (r + 1) (3 r - 1)$
which is a composite number.

Q. Let Vr​ done the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r−1). Let Tr​=Vr+1​−Vr​−2 and Qr​=Tr+1​−Tr​ for r=1,2,… Tr​ is always